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Explicit bounds for primes in residue classes

Authors:	Eric Bach Jonathan Sorenson

Institution:	Computer Sciences Department University of Wisconsin Madison, Wisconsin 53706 ; Department of Mathematics and Computer Science Butler University Indianapolis, Indiana 46208

Abstract:	Let be an abelian extension of number fields, with $E \ne \Bbb Q$ . Let $\Delta$ and denote the absolute discriminant and degree of . Let $\sigma$ denote an element of the Galois group of . We prove the following theorems, assuming the Extended Riemann Hypothesis: (1) There is a degree- prime ${\frak p}$ of such that $\left (\frac {\displaystyle \frak p}{E/K}\right ) =\sigma$ , satisfying $N{\frak p}\le (1+o(1))(\log \Delta+2n)^2$ . (2) There is a degree- prime $\frak p$ of such that $\left (\frac {\displaystyle \frak p}{E/K}\right )$ generates the same group as $\sigma$ , satisfying $N{\frak p}\le (1+o(1))(\log \Delta)^2$ . (3) For $K=\Bbb Q$ , there is a prime such that $\left (\frac {\displaystyle \frak p}{E/\Bbb Q }\right )=\sigma$ , satisfying $p\le (1+o(1))(\log \Delta)^2$ . In (1) and (2) we can in fact take $\frak p$ to be unramified in $K/\Bbb Q$ . A special case of this result is the following. (4) If $\gcd (m,q)=1$ , the least prime $p\equiv m\pmod q$ satisfies $p\le (1+o(1))(\varphi (q)\log q)^2$ . It follows from our proof that (1)--(3) also hold for arbitrary Galois extensions, provided we replace $\sigma$ by its conjugacy class $\langle \sigma\rangle$ . Our theorems lead to explicit versions of (1)--(4), including the following: the least prime $p \equiv m \pmod q$ is less than $2( q \log q )^2$ .

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